Theorem axpow3

Description: A variant of the Axiom of Power Sets ax-pow . For any set x , there exists a set y whose members are exactly the subsets of x i.e. the power set of x . Axiom Pow of BellMachover p. 466. (Contributed by NM, 4-Jun-2006)

	Ref	Expression
Assertion	axpow3	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		axpow2	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦 )
2	1	bm1.3ii	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥 )
3		bicom	⊢ ( ( 𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦 ) ↔ ( 𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥 ) )
4	3	albii	⊢ ( ∀ 𝑧 ( 𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦 ) ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥 ) )
5	4	exbii	⊢ ( ∃ 𝑦 ∀ 𝑧 ( 𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦 ) ↔ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥 ) )
6	2 5	mpbir	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦 )